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Properties

Label 2-245-35.12-c1-0-2
Degree $2$
Conductor $245$
Sign $-0.347 - 0.937i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.366i)2-s + (−0.578 + 2.15i)3-s + (−2.15 + 0.578i)5-s + 3.16i·6-s + (−1.99 + 2i)8-s + (−1.73 − i)9-s + (−2.73 + 1.58i)10-s + (0.5 + 0.866i)11-s + (1.58 + 1.58i)13-s − 5i·15-s + (−1.99 + 3.46i)16-s + (2.15 + 0.578i)17-s + (−2.73 − 0.732i)18-s + (1.58 − 2.73i)19-s + (1 + 0.999i)22-s + (0.732 + 2.73i)23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.334 + 1.24i)3-s + (−0.965 + 0.258i)5-s + 1.29i·6-s + (−0.707 + 0.707i)8-s + (−0.577 − 0.333i)9-s + (−0.866 + 0.499i)10-s + (0.150 + 0.261i)11-s + (0.438 + 0.438i)13-s − 1.29i·15-s + (−0.499 + 0.866i)16-s + (0.523 + 0.140i)17-s + (−0.643 − 0.172i)18-s + (0.362 − 0.628i)19-s + (0.213 + 0.213i)22-s + (0.152 + 0.569i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.347 - 0.937i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.347 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.743081 + 1.06813i\)
\(L(\frac12)\) \(\approx\) \(0.743081 + 1.06813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.15 - 0.578i)T \)
7 \( 1 \)
good2 \( 1 + (-1.36 + 0.366i)T + (1.73 - i)T^{2} \)
3 \( 1 + (0.578 - 2.15i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.58 - 1.58i)T + 13iT^{2} \)
17 \( 1 + (-2.15 - 0.578i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.58 + 2.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.732 - 2.73i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (-2.73 + 1.58i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.19 + 2.19i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 9.48iT - 41T^{2} \)
43 \( 1 + (3 - 3i)T - 43iT^{2} \)
47 \( 1 + (-1.73 - 6.47i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.36 + 0.366i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.74 - 8.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.47 + 3.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.366 - 1.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (11.2 + 6.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.16 - 3.16i)T + 83iT^{2} \)
89 \( 1 + (3.16 - 5.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.58 + 1.58i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.22484744706561590397631965438, −11.45145990278537822517524765669, −10.83081839501123749034619681746, −9.607065694736171129472111764540, −8.681226684352573077350922102591, −7.37366710560916473540073437251, −5.84456768143362351440127507689, −4.71948346472301104859170922475, −4.06365461765107083449186323613, −3.12939704677017520537522345696, 0.870423503902360174873686875395, 3.26168053340668380680452880917, 4.49193194109478134567999353000, 5.73914840132272136024167381639, 6.58227702065725419309292602978, 7.63824402489254727222502398383, 8.495172656422808545720629396286, 9.934324982464624059041991936846, 11.45130526196939501893658597216, 12.05984922555760572453509340524

Graph of the $Z$-function along the critical line